Theorem 0nelopabOLD 5564

Description: Obsolete version of 0nelopab 5563 as of 3-Oct-2024. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
0nelopabOLD	⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem 0nelopabOLD

Step	Hyp	Ref	Expression
1		elopab 5523	. . 3 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		nfopab1 5212	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	2	nfel2 2916	. . . . 5 ⊢ Ⅎ𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4	3	nfn 1853	. . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5		nfopab2 5213	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
6	5	nfel2 2916	. . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
7	6	nfn 1853	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
8		vex 3473	. . . . . . . 8 ⊢ 𝑥 ∈ V
9		vex 3473	. . . . . . . 8 ⊢ 𝑦 ∈ V
10	8, 9	opnzi 5470	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
11		nesym 2992	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩)
12		pm2.21 123	. . . . . . . 8 ⊢ (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
13	11, 12	sylbi 216	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
14	10, 13	ax-mp 5	. . . . . 6 ⊢ (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15	14	adantr 480	. . . . 5 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
16	7, 15	exlimi 2203	. . . 4 ⊢ (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
17	4, 16	exlimi 2203	. . 3 ⊢ (∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
18	1, 17	sylbi 216	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
19		id 22	. 2 ⊢ (¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
20	18, 19	pm2.61i 182	1 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  ∅c0 4318  ⟨cop 4630  {copab 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5205

This theorem is referenced by: (None)